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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>At the maximum height, denoted by <span class="process-math">\(\xi\text{,}\)</span> the velocity is zero:</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_24.html">
\begin{equation}
0=\pm  \sqrt{\frac{2gR^2}{R+\xi}+v_0^2-2gR}~\rightarrow~\xi=\frac{v_0^2R}{2gR-v_0^2}.\tag{2.5.8}
\end{equation}
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<p class="continuation">Not returning to the earth implies <span class="process-math">\(\xi~\rightarrow~\infty\text{.}\)</span> In (<a href="" class="xref" data-knowl="./knowl/eq2_24.html" title="Equation 2.5.8">(2.5.8)</a>), letting <span class="process-math">\(\xi~\rightarrow~\infty\text{,}\)</span> we must have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_24.html">
\begin{equation*}
2gR-v_0^2=0~\rightarrow~v_0=\sqrt{2gR},
\end{equation*}
</div>
<p class="continuation">which is the smallest initial velocity. For <span class="process-math">\(g=9.8m/s^2, R\approx 6286224.5m\text{,}\)</span> <span class="process-math">\(v_0=11100 m/s\)</span> or <span class="process-math">\(v_0=11.1 km/s\text{.}\)</span></p>
<span class="incontext"><a href="sec2_5.html#p-40" class="internal">in-context</a></span>
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